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Teaching in general and teaching relativity in particular consists
in large measure of providing a framework of ideas, notation, and
other formalism so that people can think about the topic without
risk of getting confused.
I have presented the key ideas in analogical terms, i.e. comparing clocks
to odometers. I have given the graphical representation, i.e. spacetime
diagrams. I have also given the matrix representation. I have also
given the bivector / quaternion / Clifford algebra representation. If
this doesn't meet your standards, please tell me what *would* be good
enough for you.
For thousands of years, people have considered__________________________end snip_____________________________________
the length of a ruler to be invariant with respect to rotations. The
projection of a ruler on the wall of the cave has been recognized as
not "really" a ruler, just a projection. As Joel R. says, it is "really"
a projection ... but it is not "really" a ruler.
It seems overwhelmingly probable that if people were more familiar with
rotations in the (t, x) plane -- i.e. boosts -- they would consider
the relevant property of a clock to be the invariant interval between
clock-ticks. Anybody in his right mind would want it this way, so why
not let it be this way?
To say the same thing in other words: considering just plain Euclidean
rotations, let the (x', y') frame be rotated relative to the (x, y) frame.
The tick-marks of either frame "(as viewed from the other frame)" will
appear foreshortened, but I haven't heard anybody arguing that either
frame "really" becomes shorter.